A272659 -id:A272659 - cp's OEIS frontend

A366158 Number of distinct determinants of 3 X 3 matrices with entries from {0, 1, ..., n}.

1, 5, 25, 77, 179, 355, 609, 995, 1497, 2167, 2999, 4069, 5289, 6841, 8595, 10661, 13023, 15777, 18795, 22305, 26085, 30397, 35107, 40381, 45929, 52247, 58929, 66287, 74139, 82767, 91643, 101701, 112013, 123235

Offset: 0

Robert P. P. McKone, Oct 02 2023

These determinants a(n) equivalently represent the leading coefficient (coefficient of term with degree 0) of the characteristic polynomials for such matrices, thereby providing a direct measure and lower bound of the uniqueness of these polynomials within this matrix class.

The maximal determinant counted by a(n) is A033431(n) = 2*n^3.

Robert P. P. McKone, The distinct determinants for a(0)-a(18).

Cf. A058331 (distinct determinants for 2 X 2 matrices).
Cf. A272661, A272659.
Cf. A365926.
Cf. A272658, A272660, A272662, A272663.
Cf. A033431 (maximal determinant).
Cf. A097400 (distinct consecutive entries in 3 X 3 matrix).

mat[n_Integer?Positive] := mat[n] = Array[m, {n, n}]; flatMat[n_Integer?Positive] := flatMat[n] = Flatten[mat[n]]; detMat[n_Integer?Positive] := detMat[n] = Det[mat[n]] // FullSimplify; a[d_Integer?Positive, 0] = 1; a[d_Integer?Positive, n_Integer?Positive] := a[d, n] = Length[DeleteDuplicates[Flatten[ParallelTable[Evaluate[detMat[d]], ##] & @@ Table[{flatMat[d][[i]], 0, n}, {i, 1, d^2}]]]]; Table[a[3, n], {n, 0, 9}]

from itertools import product
def A366158(n): return len({a[0]*(a[4]*a[8] - a[5]*a[7]) - a[1]*(a[3]*a[8] - a[5]*a[6]) + a[2]*(a[3]*a[7] - a[4]*a[6]) for a in product(range(n+1),repeat=9)}) # Chai Wah Wu, Oct 06 2023

a(19)-a(26) from Robin Visser, May 08 2025

a(27)-a(33) from Robin Visser, Aug 26 2025

A367978 Number of distinct characteristic polynomials for 4 X 4 matrices with entries from {0, 1, ..., n}.

Original entry on oeis.org

1, 333, 58335, 2875405, 47125558

Offset: 0

Robert P. P. McKone, Dec 07 2023

Cf. A366448 (2 X 2 matrices), A366551 (3 X 3 matrices).

Cf. A272659.

mat[n_Integer?Positive] := mat[n] = Array[m, {n, n}];
flatMat[n_Integer?Positive] := flatMat[n] = Flatten[mat[n]];
charPolyMat[n_Integer?Positive] := charPolyMat[n] = FullSimplify[CoefficientList[Expand[CharacteristicPolynomial[mat[n], x]], x]];
a[d_Integer?Positive, 0] = 1; a[d_Integer?Positive, n_Integer?Positive] := a[d, n] = Length[DeleteDuplicates[Flatten[Table[Evaluate[charPolyMat[d]], ##] & @@ Table[{flatMat[d][[i]], 0, n}, {i, 1, d^2}], d^2 - 1]]];
Table[a[4, n], {n, 0, 2}]

import itertools
def a(n):
    ans, W = set(), itertools.product(range(n+1), repeat=16)
    for w in W: ans.add(Matrix(ZZ, 4, 4, w).charpoly())
    return len(ans)  # Robin Visser, May 04 2025

a(4) from Robin Visser, May 04 2025

A306795 Number of distinct real eigenvalues of n X n matrices with elements {0, 1, 2}.

Original entry on oeis.org

3, 25, 657, 112870

Offset: 1

Steven E. Thornton, Mar 10 2019

S. E. Thornton, Properties of the Bohemian family of n x n matrices with population {0, 1, 2}, Characteristic Polynomial Database.

Number of eigenvalues is in A306792.
Number of characteristic polynomials is in A272659.
Number of minimal polynomials is in A306783.

A306818 Number of non-derogatory n X n matrices with elements {0, 1, 2}.

Original entry on oeis.org

3, 78, 19068, 42300060

Offset: 1

Steven E. Thornton, Mar 11 2019

Encyclopedia of Math, Non-derogatory matrix
S. E. Thornton, Properties of the Bohemian family of n x n matrices with population {0, 1, 2}, Characteristic Polynomial Database.

Number of characteristic polynomials is in A272659.

Number of minimal polynomials is in A306783.

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A366158 Number of distinct determinants of 3 X 3 matrices with entries from {0, 1, ..., n}.

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A367978 Number of distinct characteristic polynomials for 4 X 4 matrices with entries from {0, 1, ..., n}.

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A306795 Number of distinct real eigenvalues of n X n matrices with elements {0, 1, 2}.

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A306818 Number of non-derogatory n X n matrices with elements {0, 1, 2}.

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